As my colleague points out, it depends. Some vets won't put down any animal they believe is "healthy", some will put down any animal as requested by its owner. You should speak with your vet about your options. You may also want to speak with a dog trainer or a breed rescue before making any decisions.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.
Technically, in Florida, an animal is property. And if the owner wants to put the animal down, then they have that right (for example, it is no crime in Florida to shoot your own horse, dog, cat, etc., as long as it is not in an abusive manner). That said, many responsible vets will not agree to put an animal down for no reason other than just an owner's convenience or choice. Why are you asking the question? Are you the one wanting to put an animal down? or are you against it? If you are wanting to put the animal down, you should check with your local animal rescue centers to see if perhaps someone wants to adopt the animal or try to find a local "no-kill" shelter to send the animal to. If you are against it, there may be little you can do other than perhaps talk to the vet and/or owner and offer to adopt the animal yourself (if the owner agrees).
Hope that helps!
Please note that THESE COMMENTS ARE NOT INTENDED AS LEGAL ADVICE and are for informational purposes only. This response is not intended to create any attorney-client relationship and is only based on the limited facts given. The response might change should additional facts be learned and should not be relied on as legal advice. It is recommended that you consult with an attorney who can properly assess the situation, as well as all pertinent facts, prior to taking any action based on the foregoing statements.